Stability theory of dynamical systems involving multiple non-monotonic Lyapunov functions

  • Anthony N. Michel
  • Ling Hou

Abstract

The stability results which comprise the Direct Method of Lyapunov involve the existence of scalar-valued functions of the system state and time (called {\it Lyapunov functions}) which when evaluated along the motions of a dynamical system decrease {\it monotonically} with increasing time and approach zero as $t\to \infty$. Functions of this type are called {\it monotonic Lyapunov functions}. More recent work concerning the qualitative analysis of ``contemporary dynamical systems'' (including switching systems, impulsive dynamical systems, hybrid dynamical systems, discrete-event systems, and the like) has given rise to Lyapunov-like stability results where the requirement that the Lyapunov functions decrease monotonically along the system motions has been relaxed. Functions of this type are called {\it non-monotonic Lyapunov functions}. It has been shown that the classical stability results involving monotonic Lyapunov functions always reduce to corresponding results involving non-monotonic Lyapunov functions and in most cases, the results involving monotonic Lyapunov functions are in general more conservative than corresponding results involving non-monotonic Lyapunov functions.
In the present paper we establish results which broaden the applicability of the stability and boundedness results involving non-monotonic Lyapunov functions appreciably. These results stipulate the existence of non-monotonic Lyapunov functions which in general are dependent on the various system motions. Such function are called {\it multiple non-monotonic Lyapunov functions}. We address continuous-time finite-dimensional dynamical systems whose motions may be discontinuous. For such systems we establish results for stability, uniform stability, local and global asymptotic stability, local and global uniform asymptotic stability, local and global exponential stability, instability and complete instability of an equilibrium $x=0$. We also establish results for the uniform boundedness and the uniform ultimate boundedness of the motions of such systems. Our results constitute sufficient conditions. We demonstrate the applicability of our results by means of several specific examples.
Published
2016-02-28
Section
Articles